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Theorem exmidontri2or 7241
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
Assertion
Ref Expression
exmidontri2or  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
Distinct variable group:    x, y

Proof of Theorem exmidontri2or
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 exmidontriim 7223 . . 3  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
2 onelss 4387 . . . . . . . 8  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
32adantl 277 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  x  C_  y
) )
4 orc 712 . . . . . . 7  |-  ( x 
C_  y  ->  (
x  C_  y  \/  y  C_  x ) )
53, 4syl6 33 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  ( x  C_  y  \/  y  C_  x ) ) )
6 eqimss 3209 . . . . . . . 8  |-  ( x  =  y  ->  x  C_  y )
76, 4syl 14 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  y  \/  y  C_  x ) )
87a1i 9 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  =  y  ->  ( x  C_  y  \/  y  C_  x ) ) )
9 onelss 4387 . . . . . . . 8  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
109adantr 276 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  y  C_  x )
)
11 olc 711 . . . . . . 7  |-  ( y 
C_  x  ->  (
x  C_  y  \/  y  C_  x ) )
1210, 11syl6 33 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( x  C_  y  \/  y  C_  x ) ) )
135, 8, 123jaod 1304 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  (
x  C_  y  \/  y  C_  x ) ) )
1413ralimdva 2544 . . . 4  |-  ( x  e.  On  ->  ( A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x
)  ->  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) ) )
1514ralimia 2538 . . 3  |-  ( A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
161, 15syl 14 . 2  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
17 ontri2orexmidim 4571 . . . 4  |-  ( A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> DECID  z  =  { (/)
} )
1817adantr 276 . . 3  |-  ( ( A. x  e.  On  A. y  e.  On  (
x  C_  y  \/  y  C_  x )  /\  z  C_  { (/) } )  -> DECID 
z  =  { (/) } )
1918exmid1dc 4200 . 2  |-  ( A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> EXMID )
2016, 19impbii 126 1  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    \/ w3o 977    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   (/)c0 3422   {csn 3592  EXMIDwem 4194   Oncon0 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371
This theorem is referenced by:  onntri52  7242
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